Quantal phase factors accompanying adiabatic changes. (English) Zbl 1113.81306

Summary: A quantal system in an eigenstate, slowly transported round a circuit \(C\) by varying parameters \(R\) in its Hamiltonian \(\hat H(R)\), will acquire a geometrical phase factor \(e^{i\gamma(C)}\) in addition to the familiar dynamical phase factor. An explicit general formula for \(\gamma(C)\) is derived in terms of the spectrum and eigenstates of \(\hat H(R)\) over a surface spanning \(C\). If \(C\) lies near a degeneracy of \(\hat H\), \(\gamma(C)\) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration \(\gamma(C)\) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.


81Q70 Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory
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